The probability of the union of events is given by: $$ p(E_1 \cup E_2) = p(E_1) + p(E_2) - p(E_1 \cap E_2) $$
title: Example
What is the probability that a number randomly selected from $X = \{ n \in \mathbb N^+ | n \le 100 \}$ is divisible by either $2$ or $5$ or both (by $10$)?
- Let $E_1$ be the event that the selected number in $X$ is divisible by $2$.
- Let $E_2$ be the event that the selected number in $X$ is divisible by $5$.
- Then $E_1 \cup E_2$ is the event that it is divisible by both.
We are given that $|X| = 100, |E_1| = 50, |E_2| = 20, |E_1 \cap E_2| = 10$.
$$
\begin{aligned}
p(E_1 \cup E_2) &= \frac{50}{100} + \frac{20}{100} - \frac{10}{100} \\
&= \frac{60}{100} \\
&= \frac{3}{5} \\
&= 0.6
\end{aligned}
$$